Dynamic mode decomposition for the monodomain equation in neuromuscular system

Abstract

In this thesis, the dynamic mode decomposition (DMD) was implemented and applied to the resulting snapshots matrix of the monodomain equation in the neuromuscular system. The Hodgkin-Huxley model was used to obtain the snapshots $v_l$ and the DMD was implemented in C++ into the DiHu framework. The goal of this work was to reduce the number of dimensions of the initial snapshot data. For this, DMD calculates the Koopman operator $R$ that should lead to the next snapshots matrix, V2k ≈ RV1k-1; where V1k-1 = [v1 · · · vk-1] and V2k = [v2 · · · vk]

From the eigenvalues of R, the growth rates and frequencies can be obtained, while the DMD modes can be found in the eigenvectors. For the used method of DMD, singular value decomposition (SVD) was required. This was implemented with the subroutines by LAPACK. All other matrix operations were also implemented with subroutines by either LAPACK or BLAS. The results are promising as long as the snapshots contain higher numbers of parameters and when there are more snapshots to apply DMD on. When there aren't very many parameters and snapshots, the error can rise too high to be useful data. We also found that the variables epsilon1 and epsilon0, which play a significant role for the dimension reduction, have to be chosen carefully.